Time‐continuous and time‐discontinuous space‐time finite elements for advection‐diffusion problems

International Journal for Numerical Methods in Engineering, vol. 124, pp. 3117–3144

Abstract

Abstract We construct four variants of space‐time finite element discretizations based on linear tensor‐product and simplex‐type finite elements. The resulting discretizations are continuous in space, and continuous or discontinuous in time. In a first test run, all four methods are applied to a linear scalar advection‐diffusion model problem. Then, the convergence properties of the time‐discontinuous space‐time finite element discretizations are studied in numerical experiments. Advection velocity and diffusion coefficient are varied, such that the parabolic case of pure diffusion (heat equation), as well as, the hyperbolic case of pure advection (transport equation) are included in the study. For each model parameter set, the error at the final time is computed for spatial and temporal element lengths ranging over several orders of magnitude to allow for an individual evaluation of the methods' spatial, temporal, and space‐time accuracy. In the parabolic case, particular attention is paid to the influence of time‐dependent boundary conditions. Key findings include a spatial accuracy of second order and a temporal accuracy between second and third order. The temporal accuracy tends toward third order depending on how advection‐dominated the test case is, on the choice of the specific discretization method, and on the time‐(in)dependence and treatment of the boundary conditions. Additionally, the potential of time‐continuous simplex space‐time finite elements for heat flux computations is demonstrated with a piston ring pack test case and a subtractive manufacturing test case.

Authors 4

1. Max von Danwitz corresponding

   Universität der Bundeswehr München

   Affiliation as printed

   Institute for Mathematics and Computer‐Based Simulation (IMCS) University of the Bundeswehr Munich Neubiberg Germany

   Institute for Mathematics and Computer-Based Simulation (IMCS) University of the Bundeswehr Munich Neubiberg Germany

2. Igor Voulis

   University of Göttingen

   Affiliation as printed

   Institute for Numerical and Applied Mathematics University of Göttingen Göttingen Germany

3. Norbert Hosters Aachen Chair for Computational Analysis of Technical Systems (CATS)

   RWTH Aachen University

   Affiliation as printed

   Chair for Computational Analysis of Technical Systems (CATS) RWTH Aachen University Aachen Germany

   Chair for Computational Analysis of Technical Systems (CATS) RWTH Aachen University Aachen Germany

4. Marek Behr Aachen Chair for Computational Analysis of Technical Systems (CATS)

   RWTH Aachen University

   Affiliation as printed

   Chair for Computational Analysis of Technical Systems (CATS) RWTH Aachen University Aachen Germany

   Chair for Computational Analysis of Technical Systems (CATS) RWTH Aachen University Aachen Germany